wanminliu@gmail.com
Ext
open
boundary
¥6
-
-
-
Heroin
-
µlEy=x
The
btD
is
not
in
D.
y
=
&
yzo
Y=
-
&yzo--
A-
:
I
;¥×]󲍻
y
so
g.
ix.
YI
has
meany
on
D
.
wanminliu@gmail.com
=¥
what's
ds
¥
,
#
ds
"
if
the
syne
is
given
by
a
graph
2-
=
2-
ix.
YI
,
where
4=1
ix.
y
,
HEIR
}
/
then
ds
=
i-zi-aytdxdy.x-yi-z-l.XZta.to/'#i+y'+z'--
1
.
and
a
is
the
distant
so
we
can
whet
from
the
observer
C-
the
center
=
142-1
,
4=4
7=2
of
the
moon
.
we
use
Y
and
2-
as
parameter
so
ds=*j+x
dydz
=
-'
Sime
is
positive
¥¥É÷
.
y
=
I
(
1-
y
'
-
2-25
?
C-
4)
=
-
y
a-
yIzj±
It
z
=
÷
(
1-
y
?
2-
'
-22-1=-2-11
-
y
?zY¥
wanminliu@gmail.com
Tien
¥1
!ds=¥÷_
pqgto-YF.net
dS=ÑÉx
dydz
Next
,
we
give
a
description
of
=
.
dydt
tepwjetm
g-
Y
to
tie
7-
7-
plane
Y
=
rcoso
2-
=
rsino
o
e
re
t
-
Eso
EI
=
dyda
ñ
I
=
¥µ
aydx
=
¥1
.
1.
±
.
=
¥
.
-4
"d
.
a
0
1-
F-
u
arms
fr om
tail
9-
=
f-
fat
.
d÷=&
WE
/
=
-441
-
E)
wanminliu@gmail.com
why
dS=HZ×i-Ñdxdy
?
'
¥
-
let
the
surfer
by
Ñcx
-
y
)
=
IX.
Y
,
2-
ix.
Y
)
)
p×
Then
Ñx
1×-77
=
11,0
,
2-
x
)
@
=
angle
between
r}
IX.
Y
)
=
10
,
I
,
2-7
)
rI
and
try
*
xril-lrillriis.mil
e-
=
ri.ru?--FxlYriill-wi01
a
-
-
5.
ri
=
Kii
-
lriiryiuioe-ri.rs
=*.ñlñ•ñ
-
lñ÷÷
=
T=
I
-
So
ds
=/
Fiery
/
=dxdy=Ea
dxdy
wanminliu@gmail.com
If
ÑCX
,
7)
=
I
.
Y
,
2-
IX.
Y
)
)
then
ri
=
11
.
0
,
2-
x
)
try
=
10
,
I
,
Zy
)
so
=L
=
RI
-
RI
=
(
1,0
,
2×1
.
I
1.0
.
Zx
)
=
It
2-
I
G-
=
try
.
Try
=
I
0. 1 . 7
,
)
-110,1
,
241=11-2-5
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,&
So
-Ñ=µxx) fttyJ-lZxZ
=ltai+zi-Ézaiy#󲍻
wanminliu@gmail.com
By
this
method
,
"
"
"
"
4-
we
pick
the
surface
F-
=
rj.ro?as x--XYY.Z)G--Ir.rI
go
the
svfneitShtnbY.E@Iz-1-xyi-xaTrTY.t_lXlY.Z1.Y
,
2-
)
try
=
I
Xy
,
I
,
0
)
Pz
=
I
2-
,
0
,
I
7
wanminliu@gmail.com
Pet
R
be
the
region
bounded
by
the
compute
1St
xydv
or
by
using
the
cylinder
coordinate
cylinder
5+5=1
and
the
planes
2-
=
1
,
7=0
S¥.oylindercoodtnaÉ
in
the
font
octant
(
r
,
O
.
2-
)
coordinate
in
the
plane
f-
DV
=
ds
.
It
=
room
_#
=
"
rdrdcidz-Y-rsho-xd-rdr-rfooi.no
)
󲍻÷÷i÷"¥
<
a
wanminliu@gmail.com
=
if
&
-
I
compute
fssxydy
or
by
Gaius
divergence
Theorem
.
Gauss
divergence
theorem
.
f-
-4
!
"
"
)
¥
gggg
,
w=É*s
=
f-
E)
fast
.
󲍻
-
ciao
-
É
-
ix.
=
-1-41×1-21
É
=
I
r
s
wanminliu@gmail.com
S
=
S
,
U
Seuss
vsx
Uss
we
need
a
vertu
field
É
is
the
boon
day
not
R
.
go
that
dw
=p
=
XY
For
eah
Si
,
we
can
denote
its
unit
normal
veto
by
Ñ
,
we
can
pick
up
É=
OF
-105-1
xx
É
.
-
-
-
-
,
?
abaefdises-divf-C-x.g-y.bz
)
(0, 9×77)=117
.
So
-
in
#
-
plane
:*
:;¥ ¥ ± :¥ É÷É¥ ÷
.
am
Ifl
dived
V
=
E.
Ñds
n
s
5g
(
front
cylinder
)
S
,
-
bottom
Edina
wanminliu@gmail.com
Then
Ñ
,
=
(
0,0
,
-1
)
IffdivÉdv=ÉifÉ.Ñds
Ñ
,
=
(
o
,
o
,
1)
n
si
Ñ
,
=
I
-1
,
0,0
)
the
3rd
component
"
"
"
=
"
"
"
"
"
Ñ
NJ
=L
,
,
0
)
is
0
.
became
His
so
=P
.
Ñi
=
0
1=13,4
,
5
in
the
y
-
plane
É
.
Ñz=
Xyz
/
(
2-
=
const
)
.
For
5
,
,
=
10.0.0
)
.
2-
=/
=D
xyds
=
If
Yds
=)
!
r
>
saoaodrdo
=
f-
q
Ey
's
/
"
%
armoire
,
y=rs W
osos
E
(7--1)
wanminliu@gmail.com
¥-4k
be
the
intersection
of
thigh
theorist
Kroupa
.
.
and
origin
is
the
the
plane
9=7
and
the
sphere
center
of
the
sphere
Yy
'
-12-1-1
Sette
intersects
in
the
director
counterclockwise
is
a
big
circle
.
with
respect
to
2-
-
axis
.
1×45+2-2=1
󲍻
1×7-22-41
compute
the
integral
7=2
Y=
2-
=
WI
§
xyzdz
|
=
v0
g
,
nanny
.my
,
gun
,
,
÷?g󲍻
y
,
÷
,
,
,
µ
,
,
1-
2-
=
¥
since
It
=
¥
cow
do
compute
the
same
integral
so
§
Hdz
=/
o
"
v0
ftp.sina/Krsinojtrirod0
1-
by
stocks
's
theorem
.
=
z-f.fi
"
@
sojlsmojdo
wanminliu@gmail.com
22
=
¥
#
f.
Eggnog
.
do
¥met
We
wan t
to
use
stocks
:
Theon
5h20
we
pick
up
the
vertov
field
|
fgynzoj
=
¥40
É
=
(
o
,
o
,
XYZ
)
d
}
=L
dx.dy.dz
)
=
¥1
.
"
;
so
so
DE
=
Xyz
dz
=
*
.
£
.
2k
=
a
§
.
di
=
If
E.
Ñds
1-
9
s
stocks
's
Theorem
wanminliu@gmail.com
So
Ñ=
10
.
¥
)
where
the
boundy
of
the
Sunfire
S
is
the
closet
yxÉ=anlÉ
ii.
I
curve
T
.
i
sufo
S
/
o
,
o
,
XYZ
s
"
*
=/
÷
:|
0*+14-12=0
=
2K¥
T
-
3×1×42-1
'T
a
normal
dtvatn
{
is
(
0
,
I
,
-
1)
=
IXZ
,
-77
,
0
)
the
plane
2=4
or
10
,
-1
,
a)
So
E)
Ñ
we
need
the
3-
rdumpwtt
be
2=1× 7
,
-47,0110
,
-
¥
,
¥
)
polka
,
so
we
pick
up
Ñ=
=
7¥
4
unit
normal
vertu
feud
wanminliu@gmail.com
I
}
E)
Ñds
=
If
as
ñ+y
:-c
7=2
S
The
plane
z=y
.
SIT
'd×dyµ÷w Y,¥
X'
+
visa
dS=HZ×4 Ñ=dxdy
=
,+ d×q= rÉ""¥
"
am
§¥ds=ss÷
.
,
÷÷÷÷÷÷
=
a
¥¥